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f : S \to (0,1) with $f(S_1 \cup S_2) = f(S_1)+ f(S_2), from set of polygonals

Source: Romania IMO TST 1991 p4

February 19, 2020
functioncombinatorial geometrygeometry

Problem Statement

Let SS be the set of all polygonal areas in a plane. Prove that there is a function f:S→(0,1)f : S \to (0,1) which satisfies f(S1∪S2)=f(S1)+f(S2)f(S_1 \cup S_2) = f(S_1)+ f(S_2) for any S1,S2∈SS_1,S_2 \in S which have common points only on their borders
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